## changing time intervals.docx - Section 2: Investigation and New Learning

# Different Time Intervals and Exponential Functions

Lesson 21 of 26

## Objective: SWBAT write exponential functions to model growth and decay using different units of time. Students will be able to change either the multiplier or the coefficient in the exponent to change the units of time in an exponential function.

#### Warm-Up

*30 min*

The first two problems are basic review of the key skills that students should understand thoroughly by this point. I make this clear to them by stating it explicitly and I reinforce this by circulating to make sure that each student works on these two problems.

The third problem is important for previewing today’s lesson. The idea is to highlight the fact that both situations have the same rate of increase, but that it happens over a different interval of time. The big question is how does this affect the function, if we want to use the same units for both functions. Students might write the exact same function for both situations, which reveals the key misconception. Ask them, “How does the number of years affect the equation? Do these two different tuitions increase at the same rate?”

The question about multipliers is important. Students will likely say that the two functions have the same multiplier.

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#### Sharing and Closing

*10 min*

The purpose of this closing is, as always to make sure that students have made sense of the key idea of this lesson, which is that the units of time matter when setting up the function and that there are two different ways to deal with this. The three possible functions are presented in the exit ticket to highlight these ideas.

Even if students did not master these new ideas during the lesson, they can figure some stuff out during the closing, if they understand how to check some data points. I start this lesson closing by asking students to tell me *how* they can check to see whether the functions fit this situation. We start by creating a quick class data table based on the initial function, with some simple input-output pairs, like and if the units of time are hours. Students can then plug these pairs into the given functions to verify that they work.

The third function was created by finding the 24^{th} root of 2 to get the new multiplier. Students might or might not fully understand this today, but at least they will realize that we can change the multiplier. If students are ready for this, you can discuss it in more detail or if a few students figure it out you can have them explain it to other students.

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- UNIT 1: Linear and Nonlinear Functions
- UNIT 2: Piecewise Functions
- UNIT 3: Absolute Value Functions and More Piecewise Functions
- UNIT 4: Introduction to Quadratic Functions through Applications
- UNIT 5: More Abstract Work with Quadratic Functions
- UNIT 6: Rational Functions
- UNIT 7: Polynomial Functions
- UNIT 8: Exponential Functions
- UNIT 9: Ferris Wheels
- UNIT 10: Circles
- UNIT 11: Radical Functions
- UNIT 12: Cubic Functions

- LESSON 1: Bunnies and Exponential Growth
- LESSON 2: More Bunnies and Exponential Growth
- LESSON 3: Candy Bars and Exponential Decay
- LESSON 4: Bunnies and Exponential Equations
- LESSON 5: Graphing Bunnies
- LESSON 6: Exponential Data Tables
- LESSON 7: Fitting Exponential Functions Given Two Points
- LESSON 8: Matching Exponential Graphs to Equations
- LESSON 9: Exponential Functions Review
- LESSON 10: Exponential Functions Portfolio and Summative Assessment
- LESSON 11: Exponential Functions and Approach Statements
- LESSON 12: Graphing Exponential Functions
- LESSON 13: Matching Graphs of Exponential Functions to their Equations
- LESSON 14: Exponential Function Designs
- LESSON 15: Graph Exponential Functions Review
- LESSON 16: Graph Exponential Functions Summative Assessment and Portfolio
- LESSON 17: Bouncy Ball Investigation
- LESSON 18: Percent Change: Growth and Decay
- LESSON 19: More Percent Changes and Exponential Functions
- LESSON 20: Writing Exponential Functions to Solve Problems
- LESSON 21: Different Time Intervals and Exponential Functions
- LESSON 22: Compound Interest
- LESSON 23: Compound Interest Formula
- LESSON 24: Continuously Compounded Interest
- LESSON 25: Applications of Exponential Functions Review
- LESSON 26: Applications of Exponential Functions Summative Assessment